This study uses an Elzaki decomposition method with two fractional derivatives to solve a fractional nonlinear coupled system of Whitham-Broer-Kaup equations. For the fractional derivatives, we used Caputo and Atangana-Baleanu derivatives in the Caputo manner. Furthermore, the proposed techniques are compared to the solutions of other renowned analytical methods, including the Adomian decomposition technique, variation iteration technique, and homotopy perturbation technique. We used two nonlinear problems to illustrate the accuracy and validity of the proposed approaches. The results of numerical simulations were used to verify that the proposed methods are accurate and efficient, and the results are displayed in graphs and tables. The obtained results demonstrate that the algorithm is very real, simple to apply, and effective in investigating the nature of complicated nonlinear models in science and engineering.